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Comparing distributions by multiple testing across quantiles or CDF values

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  • Matt Goldman
  • David M. Kaplan

Abstract

When comparing two distributions, it is often helpful to learn at which quantiles or values there is a statistically significant difference. This provides more information than the binary "reject" or "do not reject" decision of a global goodness-of-fit test. Framing our question as multiple testing across the continuum of quantiles $\tau\in(0,1)$ or values $r\in\mathbb{R}$, we show that the Kolmogorov--Smirnov test (interpreted as a multiple testing procedure) achieves strong control of the familywise error rate. However, its well-known flaw of low sensitivity in the tails remains. We provide an alternative method that retains such strong control of familywise error rate while also having even sensitivity, i.e., equal pointwise type I error rates at each of $n\to\infty$ order statistics across the distribution. Our one-sample method computes instantly, using our new formula that also instantly computes goodness-of-fit $p$-values and uniform confidence bands. To improve power, we also propose stepdown and pre-test procedures that maintain control of the asymptotic familywise error rate. One-sample and two-sample cases are considered, as well as extensions to regression discontinuity designs and conditional distributions. Simulations, empirical examples, and code are provided.

Suggested Citation

  • Matt Goldman & David M. Kaplan, 2017. "Comparing distributions by multiple testing across quantiles or CDF values," Papers 1708.04658, arXiv.org.
  • Handle: RePEc:arx:papers:1708.04658
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    References listed on IDEAS

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    Cited by:

    1. Goldman, Matt & Kaplan, David M., 2017. "Fractional order statistic approximation for nonparametric conditional quantile inference," Journal of Econometrics, Elsevier, vol. 196(2), pages 331-346.
    2. David M. Kaplan & Matt Goldman, 2015. "Nonparametric inference on conditional quantile differences and linear combinations, using L-statistics," Working Papers 1503, Department of Economics, University of Missouri.
    3. Matt Goldman & David M. Kaplan, 2018. "Non‐parametric inference on (conditional) quantile differences and interquantile ranges, using L‐statistics," Econometrics Journal, Royal Economic Society, vol. 21(2), pages 136-169, June.
    4. David M. Kaplan & Longhao Zhuo, 2016. "Frequentist size of Bayesian inequality tests," Papers 1607.00393, arXiv.org, revised Feb 2018.
    5. David M. Kaplan & Longhao Zhuo, 2015. "Frequentist properties of Bayesian inequality tests," Working Papers 1910, Department of Economics, University of Missouri, revised Jul 2019.
    6. John Mullahy, 2020. "Discovering Treatment Effectiveness via Median Treatment Effects—Applications to COVID-19 Clinical Trials," NBER Working Papers 27895, National Bureau of Economic Research, Inc.
    7. Klenio Barbosa & Dakshina De Silva & Liyu Yang & Hisayuki Yoshimoto, 2020. "Bond Losses and Systemic Risk," Working Papers 288072615, Lancaster University Management School, Economics Department.
    8. Fredrik Heyman & Pehr-Johan Norbäck & Lars Persson, 2020. "Talent, Career Choice and Competition: The Gender Wage Gap at the Top," CESifo Working Paper Series 8657, CESifo.
    9. David M. Kaplan, 2020. "Inference on Consensus Ranking of Distributions," Working Papers 2010, Department of Economics, University of Missouri.

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    More about this item

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models

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